We derive a Sequential Minimal Optimization (SMO) algorithm for the quadratic dual problem arising from $\varepsilon$-SVR~\cite{Vapnik1995, Drucker1997, Smola2004} modified to minimize the Mean Absolute Percentage Error (MAPE)~\cite{Makridakis1993, Hyndman2006} directly in the loss function~\cite{benavides2025support}. This formulation is part of a broader family of SVR models with percentage-error losses that also includes least-squares variants~\cite{Suykens2002} and symmetric-kernel extensions~\cite{Espinoza2005}, whose unified structure is studied in~\cite{benavides2026unified}. The key structural difference from standard $\varepsilon$-SVR is that the box constraints become \emph{sample-dependent}: $α_k, α_k^* \in [0,\, 100C/y_k]$. We show that this modification affects only (i) the feasibility sets $\Iup$ and $\Idown$ in the working-set selection and (ii) the clipping bounds in the analytic two-variable update, while leaving the curvature formula and gradient update structurally identical to the standard SMO~\cite{Platt1998, Platt1999, Fan2005}. A shrinking heuristic adapted to the sample-dependent bounds is derived and shown to introduce an asymmetry between $α$- and $α^*$-variables controlled by the gap $2y_k\varepsilon/100$. The same solver applies to the symmetric-kernel variant (m2) by replacing $Ω$ with $Ω_s = \tfrac{1}{2}(Ω+ aΩ^*)$~\cite{Espinoza2005}. Numerical validation against an interior-point QP reference solver confirms solution agreement to within solver termination tolerance across ten synthetic configurations spanning both kernel variants and symmetry types. An implementation is available in the open-source \texttt{psvr} R package~\cite{BenavidesHerrera2026Rpsvr}.

We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors. Our test avoids the need to maximize the possibly multimodal likelihood function of such models and is applicable to models with larger number of variables. These points are illustrated in numerical experiments.

Recent advances in constrained reinforcement learning (RL) have endowed reinforcement learning with certain safety guarantees.

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Given training data ofn-dimensional LPs, we learn ann k projection matrix withn>k.

The classical optimal transport (OT) problem seeks the map that moves mass from a source to a target measure while minimizing a prescribed cost function. The objective can be formalized in either Monge's [12] or Kantronich's formulation [10], a convex relaxation of the former that considers transport plans instead of deterministic maps. These foundational formulations have wide-ranging applications, including to economics [7] and machine learning [14]. In many practical scenarios, the source measure is known or readily in-ferrable from empirical data but the target measure is not explicitly specified. Instead, it is only constrained by practical requirements or expert knowledge. For example, when applying Monge's formulation to transportation problems, the placement of the mass in the target region may be constrained to lie entirely beyond a certain boundary or within a particular region, rather than by the specification of a precise location for each fraction of the total mass. Similarly, in economic applications, supply and demand may be subject to constraints such as maximal amounts available or minimal amounts required, rather than dictated through precise marginal distributions. 1

Speculative sampling reduces the latency of autoregressive decoding for target model LLMs without sacrificing inference quality, by using a cheap draft model to suggest a candidate token and a verification criterion to accept or resample this token. To improve acceptance and decoding efficiency, recent work has explored the multi-draft extension, where at each step $n$ draft tokens are generated, and the verification criterion is a distribution conditioned on these. When this criterion maximizes the probability of accepting some draft token, it is called the optimal transport (OT). However, finding the OT is difficult, as it is the solution of a linear program (OTLP) in over $V^n$ variables, with $V$ being the vocabulary size. Two recent theoretical works have reframed the OTLP in terms of importance sampling or subset selection. In this work, we prove that these formulations are equivalent to an exponentially large relaxed OTLP, so it remains infeasible to solve. Then, we reverse engineer subset selection to formulate the OTLP as a max-flow problem. With a novel application of polymatroid theory, we reduce the exponentially large OTLP to a convex optimization problem in at most $V$ variables. This allows us to devise an algorithm for optimal $n$-draft speculative sampling when the $n$ tokens are chosen i.i.d. from a single draft model, which can be tuned to arbitrary accuracy. Finally, we measure acceptance rates and algorithm runtimes for various $n$ and top-$k$ draft sampling settings. Our findings give the first multi-draft algorithm with 90% acceptance and under 100 ms of overhead per generated token with negligible deviation from the target model distribution.